Special Cases With No Solutions
An Apollonius problem is impossible if the given circles are nested, i.e., if one circle is completely enclosed within a particular circle and the remaining circle is completely excluded. This follows because any solution circle would have to cross over the middle circle to move from its tangency to the inner circle to its tangency with the outer circle. This general result has several special cases when the given circles are shrunk to points (zero radius) or expanded to straight lines (infinite radius). For example, the CCL problem has zero solutions if the two circles are on opposite sides of the line since, in that case, any solution circle would have to cross the given line non-tangentially to go from the tangent point of one circle to that of the other.
Read more about this topic: Special Cases Of Apollonius' Problem
Famous quotes containing the words special, cases and/or solutions:
“I have a special grudge against those who have the same faults as I do.”
—Mason Cooley (b. 1927)
“I do not believe in lawyers, in that mode of attacking or defending a man, because you descend to meet the judge on his own ground, and, in cases of the highest importance, it is of no consequence whether a man breaks a human law or not. Let lawyers decide trivial cases.”
—Henry David Thoreau (1817–1862)
“Those great ideas which come to you in your sleep just before you awake in morning, those solutions to the world’s problems which, in the light of day, turn out to be duds of the puniest order, couldn’t they be put to some use, after all?”
—Robert Benchley (1889–1945)